Lusztig, George Quiver varieties and Weyl group actions. (English) Zbl 0958.20036 Ann. Inst. Fourier 50, No. 2, 461-489 (2000). For a finite graph of type \(ADE\) with set of vertices \(I\), H. Nakajima associated to \(\mathbf{v,w}\in\mathbb{N}^I\) a “quiver variety” \(\mathcal M(\mathbf{v,w})\) and constructed a Weyl group action on the cohomology of \(\sqcup_{\mathbf v}\mathcal M(\mathbf{v,w})\) using techniques of hyper-Kähler geometry [see Duke Math. J. 76, No. 2, 365-416 (1994; Zbl 0826.17026), Duke J. Math. 91, No. 3, 515-560 (1998)]. In the paper under review it is given an alternative construction of this Weyl group action, based not on hyper-Kähler geometry, but on techniques of intersection cohomology. This gives in fact a refinement of the Weyl group action. Reviewer: Vladimir L.Popov (Wien) Cited in 2 ReviewsCited in 13 Documents MSC: 20G10 Cohomology theory for linear algebraic groups 20G05 Representation theory for linear algebraic groups 14L30 Group actions on varieties or schemes (quotients) Keywords:quiver varieties; Weyl groups; intersection cohomology; Weyl group actions PDF BibTeX XML Cite \textit{G. Lusztig}, Ann. Inst. Fourier 50, No. 2, 461--489 (2000; Zbl 0958.20036) Full Text: DOI Numdam EuDML References: [1] G. LUSZTIG, Green polynomials and singularities of unipotent classes, Adv. in Math., 42 (1981), 169-178. · Zbl 0473.20029 [2] G. LUSZTIG, Cuspidal local systems and representations of graded Hecke algebras, I, Publ. Math. IHES, 67 (1988), 145-202. · Zbl 0699.22026 [3] G. LUSZTIG, Quivers, perverse sheaves and enveloping algebras, J. Amer. Math. Soc., 4 (1991), 365-421. · Zbl 0738.17011 [4] G. LUSZTIG, On quiver varieties, Adv. in Math., 136 (1998), 141-182. · Zbl 0915.17008 [5] H. NAKAJIMA, Instantons on ALE spaces, quiver varieties and Kac-Moody algebras, Duke J. Math., 76 (1994), 365-416. · Zbl 0826.17026 [6] H. NAKAJIMA, Quiver varieties and Kac-Moody algebras, Duke J. Math., 91 (1998), 515-560. · Zbl 0970.17017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.